Exceptional–point–enhanced phase sensing

Optical sensors, crucial in diverse fields like gravitational wave detection, biomedical imaging, and structural health monitoring, rely on optical phase to convey valuable information. Enhancing sensitivity is important for detecting weak signals. Exceptional points (EPs), identified in non-Hermitian systems, offer great potential for advanced sensors, given their marked response to perturbations. However, strict physical requirements for operating a sensor at EPs limit broader applications. Here, we introduce an EP-enhanced sensing platform featuring plug-in external sensors separated from an EP control unit. EPs are achieved without modifying the sensor, solely through control-unit adjustments. This configuration converts and amplifies optical phase changes into quantifiable spectral features. By separating sensing and control functions, we expand the applicability of EP enhancement to various conventional sensors. As a proof-of-concept, we demonstrate a sixfold reduction in the detection limit of fiber-optic strain sensing using this configuration. This work establishes a universal platform for applying EP enhancement to diverse phase-dependent structures, promising ultrahigh-sensitivity sensing across various applications.


Supplementary Notes 1. Theoretical analysis of splitting
In the Methods section, we have solved the eigenvalues of the two-mode-coupled system as in which  ̃ is the complex bidirectional coupling strength and  ̃ is the complex unidirectional coupling strength (see Fig. 1 in the main text).The spectral characteristics, frequency splitting and linewidth difference, are defined as 2|Ω|/2 and 2|Γ|/2, respectively.We plot them in fig.S1 at the coupling strengths / = 0.5, 1.0, and 1.5, varying the phase ∆.Both frequency splitting and linewidth difference reach zero simultaneously at the EP, where the eigenstates coalesce.As perturbed by a small phase ∆, the change of splitting around the EP state (/ = 1) is much larger than that at other states, showing EP-enhanced phase sensing.Note that a more remarkable EP enhancement is observed as perturbed by smaller phase changes (fig.S2), while the superiority of EPs degrades for larger perturbations, for example, ∆ = 1.
The symmetry of the splitting curves is affected by the parameter  (fig.S3).When the bidirectional coupling strength  ̃ is purely real, i.e.,  = 0, the curves are symmetric about the yaxis (∆ = 0).However, the dissipative term of the coupling cannot be avoided since the scatterer also acts as a loss channel, leading to non-zero .Our experimental results (Fig. 3 in the main text) display a slight asymmetry of the fitted curves.We have derived the splitting with ∆ ≪ 1 in the Methods section,  ± ≈ ± (−/4) √∆.Assuming ∆ > 0, the frequency splitting, i.e., the real part of  ± , is given as Theoretically, the maximum response to a small phase perturbation occurs at  = /4.The EP enhancement can be further improved |cos(0)| |cos(−/4)| ⁄ = √2 ≈ 1.4 times than the current results.One may tune the  by a chromium-coated nanotip (62).

Reflection-type remote strain sensor
As described in the main text, the reflection-type sensor (RS) is constructed by a fiber stretched by a piezo (PZ) stage and a fiber-based mirror (FBM).The length of stretched fiber  is set as 8 cm.The movement of the PZ stage, driven by an arbitrary waveform generator and an amplifier, leads to an optical change induced by the fiber strain.The phase change is accumulated twice due to the reflection of the FBM.To avoid unwanted mechanical instability, all the components are fixed inside an acrylic box that is placed on a suspended optical table.
To compare with our EP-enhanced sensing, we first characterize the RS is using a Mach-Zehnder interferometer (MZI).The input power is the same as the probe power (2.5 μW) used in the EP sensing experiment.An optical circulator helps to connect the RS to one arm (fig.S4B).The output voltages of the photodetector (PD) are recorded as the strain varies in fig.S4C.The electric noise through a 20-MHz low-pass filter is 2.45 mV.The detection limit is derived to be 57 nε.
The response of the EP sensing system is shown in fig.S5, as perturbed by a pulsed strain, which has a period of 2 s, a duty cycle of 50%, and an amplitude of 1.32 με.A driving voltage is applied to the PZ stage to generate the fiber strain.Replacing the fragile tapered fibers with on-chip waveguides can reduce the splitting fluctuations caused by mechanical instability, thereby improving the detection limit of the EP sensing system.
The advantage of EP states, i.e., higher sensitivity, lies in their susceptibility to small phase changes.Fig. 4C in the main text shows that the frequency splitting at an EP is larger than that in the case of  > , with the strain perturbation 0.16 με.However, for larger perturbations, for example 0.33 με, the magnitude of splitting at the EP ( = ) is very close to that at the non-EP ( > ) (fig.S6).

Transmission-type remote strain sensor
The transmission-type sensor (TS) is based on a fiber ring resonator, constructed from a 1:1 fiber splitter with the two interlinked ports externally connected.The theoretical expressions can be derived from the matrix method ( 63) where  is the amplitude ratio of the beam splitter,  is the inner circulator factor,  = / represents the phase accumulation per circle, and  is the length of the fiber resonator.In our experiments, the fiber ring resonator works in the over-coupling regime to reduce the transmission variation with changing detuning (59).The transmission of the TS affects the unidirectional coupling strength , a critical parameter for EP tuning.The approach to introducing the fiber strain is the same as described in note S2.
The TS is characterized by an MZI, which consists of three fiber splitters and one fiber combiner (fig.S7B).The transmission and the phase response of the TS are monitored by PD1 and PD2, respectively, while PD3 is used to check the reference beam.fig.S7C displays the measured transmission and the phase change.The curve fitting gives  = 0.67,  = 0.98, and  = 0.56 m.The phase change is obtained from the interference pattern.The maximum slope is about 4 times larger than the phase change without the resonant structure (dashed line).The interference patterns (fig.S7D) vary with different phases of the reference beam (−  ).The detection limit (25 nε) is derived from the noise level and estimated at the maximum slope at   ≈ /2.

Noise and mitigation strategies
In the main text, we have demonstrated the reduction of the detection limit by operating a sensor at EPs.The theoretical analysis reveals that the frequency splitting around a second-order EP scales as √ and the enhancement factor 1 √ ⁄ tends to infinity for a small perturbation  → 0. However, EP-enhanced fiber strain sensing demonstrates a finite reduction in the detection limit.The detection limit of this EP platform is determined by the fluctuation of splitting, as quantified by SNR = 1.These experiments in classical regimes are far from the fundamental noise limitation caused by non-Hermitian eigen-basis collapse.Technical noise in classical systems, such as the vibration of optical and electric measurement components, dominates in monitoring the fluctuations of splitting at transmission spectra.Below are examples of sources that may impact the detection limit.
1) Mechanical instability that deviates the system from EPs: Examples include airflow and the vibration of the optical table affecting the gap between the tapered fibers and the microtoroid (resonator-IWG coupling ), as well as the unwanted phase change in the unfixed fiber that connects the control and sensing unit ( 0 ).Thermal drifts of the piezoelectric components in the translation stages or the phase shifter are also a source of mechanical instability.
2) Thermo-refractive noise in high-quality-factor (Q) microresonators: This can be suppressed by reducing optical powers and using a medium with compensated thermo-optic coefficients (64).
3) Drift in the tunable laser output: Variations in power and center wavelength can cause this.
4) Electric noise of the photodetector and the oscilloscope: This is particularly noticeable in the case of weak optical probe signals (several microwatts to nanowatts level) and can be mitigated by improving the photodetector responsivity or applying a low-pass filter to electrical signals.
5) Thermal fluctuations: These affect the resonant frequency of the microresonator or the phase offset of IWG ( 0 ) and can be reduced by placing the components with a high-precise closedloop temperature control unit.
6) Extraction errors in the case of small frequency splitting: The small frequency splitting may be overwhelmed by broad mode linewidths so that it cannot be resolved from the transmission spectra directly.Optical gain can be introduced by erbium-ion doping to compensate for the system dissipation, as described in Materials and Methods of the main text.Additionally, the accuracy and reliability of extracting tiny splitting (e.g., sub-MHz) can be further improved by optimizing the curve fitting algorithm or with the assistance of machine-learning models.
Note that the robustness analysis (65) suggests that the EP system can be stabilized against weak parametric noise.Classical-noise-induced instability can, to some extent, be mitigated through technical methods, such as closed-loop control for drift compensation, lock-in amplifier for extracting signal obscured by various noises, and mechanically fixing fragile photonic structures.Despite the observed increase in noise at EPs, improved SNRs have been experimentally demonstrated in micromechanical (66) and electronic (27) systems.
In our experiments, the splitting fluctuation is dominated by mechanical instability.To minimize this fluctuation, we adopt the following strategies: a) The whole system is built on an actively damped optical table with pneumatic isolation; b) the EP control and sensing components are housed inside an acrylic box that is closed during experiments; c) two closed-loop piezo-electric nano-translation stages are used to precisely control the resonator-IWG coupling  and the resonator-BWG coupling; and d) the IWG fiber is fixed on the optical table to decrease the fluctuation of  0 .
Looking ahead, EP enhancement can be further improved through integrated photonics techniques, where all the optical components are fabricated on the same chip to reduce mechanical instability.Besides, coupling strengths and phases can be precisely tuned by various mechanisms, such as the electro-optic (EO) effect of lithium niobate, the thermo-optic effect of silicon nitride, and the piezoelectric effect of aluminum nitride.For example, a variable coupling strength  can be realized using a pair of 50:50 directional couplers and EO tunable optical paths, as demonstrated in Refs.(60,67).Dynamic control of EP states with stability and reliability can be achieved through high-speed EO modulation and feedback loops.

Fig. S3 .
Fig. S3.Changes of the frequency splitting (solid lines) and linewidth difference (dashed lines) as the parameter  varies.The curves are symmetric about the y-axis in the case of  = 0.With increasing , the curves become asymmetric, and the change of frequency splitting at a sufficiently small phase perturbation (0 < ∆ ≪ 1) also increases until reaching the maximum at  = /4.Parameters:  =  = 2 × 5 MHz,  0 =  + .

Fig. S4 .
Fig. S4.Characterization of the reflection-type sensor.(A) Schematic of the RS for strain sensing.(B) Characterization of the RS with an MZI.The RS is connected to one arm by an optical circulator.(C) Measured signals as the applied strain increases when the reference phase   is locked at /2.RS, reflection-type sensor; PZ, piezo; FBM, fiber-based mirror; PD, photodetector.

Fig. S5 .
Fig.S5.Frequency splitting in response to a pulsed strain.(A) The system at an EP state is perturbed by a 2s 50% pulsed strain in the fiber (1.32 με).(B) Changes in frequency splitting when the system is perturbed by a pulsed strain.The fluctuations of the frequency splitting without and with strain in the fiber are 0.32 MHz and 0.31 MHz, respectively.

Fig. S6 .
Fig.S6.Frequency splitting in response to a stronger pulsed strain.The amplitude of strains is 0.33 με.From top to bottom, the system is at the states  <  ,  =  (EP), and  >  , respectively.The EP enhancement is negligible for large perturbations.

Fig. S7 .
Fig. S7.Characterization of the transmission-type resonant strain sensor using an MZI.(A) Schematic of the fiber ring resonator for strain sensing.(B) Characterization of the TS using an MZI.The transmission (PD1) and phase (PD2) of the light through the TS, along with the transmission of the reference path (PD3), are monitored.(C) The measured transmission and phase change of the TS.Dashed line: the phase change without the resonant structure.(D) Signal output of PD2.Fitted   / = 0.04, 0.48, 0.98, respectively.TS, transmission-type sensor; PZ, piezo; PD, photodetector.

Fig. S8 .
Fig. S8.Experimental setup for the chirality measurement.When inputting the CW mode, the transmission spectra are monitored by PD2, and the intensities of CCW and CW modes are measured by PD1 and PD3, respectively.For the CCW input direction, PD1 is used to monitor the transmission spectra, while both PD2 and PD3 measure the intensity of CW modes.PD, photodetector; PS, phase shifter; FBM, fiber-based mirror; OS, optical switch.

Fig. S10 .
Fig. S10.Experimental setup for splitting measurements.The active cavity, pumped by the 1460-band light, provides optical gain for linewidth narrowing in the 1550-band probe light.PD1 and PD2 monitor the transmission spectra of the probe light and the pump light, respectively.The EP states are confirmed by zero reflections at PD3.The PS adjusts the phase offset  0 to realize EPs and compensate for additional frequency-related phase changes (Methods and fig.S14).Through the manual OS, port 3 of the control unit connects either to the FBM for the splitting characterization or to the remote sensor for sensing experiments.WFG, waveform generator; PS, phase shifter; OS, optical switch; FBM, fiber-based mirror; OSA, optical spectrum analyzer; WDM, wavelength-division multiplexing; PD, photodetector; PM, power meter; RS, reflectiontype sensor; TS, transmission-type sensor.

Fig. S11 .
Fig. S11.Transmission and reflection spectra while tuning around an EP.From bottom to top, ∆ varies from − to .The EP state exhibits an unsplit transmission spectrum and a zero reflection.

Fig. S13 .
Fig.S13.Transmission spectra with increasing probe power.The probe light in the active cavity opens a lossy channel for the gain provided by the pump(58).Therefore, higher probe power compromises the ability to use optical gain to narrow the linewidth.Inset: Broader Lorentzian lineshape of the probe light without the pump light.Our system works in the undercoupling regime.

Fig. S14 .
Fig. S14.Compensation of additional phase change induced by frequency scanning.(A) Without the modulation signal, the EP state is only achieved at zero detuning, which is confirmed by the zero reflection.The non-zero reflection elsewhere indicates the deviation from the EP state, because the frequency detuning leads to additional phase change (Materials and Methods).Also, the lineshape also deviates from the theoretical expectation.(B) With the modulation signal (4.98 kHz), the effect of the frequency-induced phase change is suppressed.The reflection remains zero at different detuning, and the lineshape of the transmission spectrum is recovered.